Exercises for Elementary Differential Geometry

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Transcript of Exercises for Elementary Differential Geometry

Exercises for

Elementary Differential Geometry

Chapter 1

1.1.1 Is (t) = (t2, t4) a parametrization of the parabola y = x2?

1.1.2 Find parametrizations of the following level curves:(i) y2 x2 = 1.(ii) x


4 +y2

9 = 1.

1.1.3 Find the Cartesian equations of the following parametrized curves:(i) (t) = (cos2 t, sin2 t).(ii) (t) = (et, t2).

1.1.4 Calculate the tangent vectors of the curves in Exercise 1.1.3.

1.1.5 Sketch the astroid in Example 1.1.4. Calculate its tangent vector at each point.At which points is the tangent vector zero ?

1.1.6 Consider the ellipsex2


q2= 1,

where p > q > 0. The eccentricity of the ellipse is =

1 q2p2 and the points(p, 0) on the x-axis are called the foci of the ellipse, which we denote by fff1 andfff2. Verify that (t) = (p cos t, q sin t) is a parametrization of the ellipse. Provethat:(i) The sum of the distances from fff1 and fff2 to any point ppp on the ellipse doesnot depend on ppp.(ii) The product of the distances from fff1 and fff2 to the tangent line at any pointppp of the ellipse does not depend on ppp.(iii) If ppp is any point on the ellipse, the line joining fff1 and ppp and that joining fff2and ppp make equal angles with the tangent line to the ellipse at ppp.

1.1.7 A cycloid is the plane curve traced out by a point on the circumference of a circleas it rolls without slipping along a straight line. Show that, if the straight lineis the x-axis and the circle has radius a > 0, the cycloid can be parametrized as

(t) = a(t sin t, 1 cos t).

1.1.8 Show that (t) = (cos2 t 12 , sin t cos t, sin t) is a parametrization of the curveof intersection of the circular cylinder of radius 12 and axis the z-axis with the



sphere of radius 1 and centre (12 , 0, 0). This is called Vivianis Curve - seebelow.

1.1.9 The normal line to a curve at a point ppp is the straight line passing through pppperpendicular to the tangent line at ppp. Find the tangent and normal lines tothe curve (t) = (2 cos t cos 2t, 2 sin t sin 2t) at the point corresponding tot = /4.

1.1.10 Find parametrizations of the following level curves:(i) y2 = x2(x2 1).(ii) x3 + y3 = 3xy (the folium of Descartes).

1.1.11 Find the Cartesian equations of the following parametrized curves:(i) (t) = (1 + cos t, sin t(1 + cos t)).(ii) (t) = (t2 + t3, t3 + t4).

1.1.12 Calculate the tangent vectors of the curves in Exercise 1.1.11. For each curve,determine at which point(s) the tangent vector vanishes.

1.1.13 If P is any point on the circle C in the xy-plane of radius a > 0 and centre (0, a),let the straight line through the origin and P intersect the line y = 2a at Q, andlet the line through P parallel to the x-axis intersect the line through Q parallelto the y-axis at R. As P moves around C, R traces out a curve called the witchof Agnesi. For this curve, find(i) a parametrization;(ii) its Cartesian equation.


1.1.14 Generalize Exercise 1.1.7 by finding parametrizations of an epicycloid (resp.hypocycloid), the curve traced out by a point on the circumference of a circle


as it rolls without slipping around the outside (resp. inside) of a fixed circle.

1.1.15 For the logarithmic spiral (t) = (et cos t, et sin t), show that the angle between(t) and the tangent vector at (t) is independent of t. (There is a picture ofthe logarithmic spiral in Example 1.2.2.)

1.1.16 Show that all the normal lines to the curve

(t) = (cos t+ t sin t, sin t t cos t)

are the same distance from the origin.

1.2.1 Calculate the arc-length of the catenary (t) = (t, cosh t) starting at the point(0, 1). This curve has the shape of a heavy chain suspended at its ends - seeExercise 2.2.4.

1.2.2 Show that the following curves are unit-speed:

(i) (t) =(

13(1 + t)

3/2, 13 (1 t)3/2, t2).

(ii) (t) =(


cos t, 1 sin t,35

cos t).

1.2.3 A plane curve is given by

() = (r cos , r sin ),

where r is a smooth function of (so that (r, ) are the polar coordinates of()). Under what conditions is regular? Find all functions r() for which is unit-speed. Show that, if is unit-speed, the image of is a circle; what is itsradius?

1.2.4 This exercise shows that a straight line is the shortest curve joining two givenpoints. Let ppp and qqq be the two points, and let be a curve passing throughboth, say (a) = ppp, (b) = qqq, where a < b. Show that, if uuu is any unit vector,


and deduce that

(qqq ppp)...uuu b



By taking uuu = (qqq ppp)/ qqq ppp , show that the length of the part of betweenppp and qqq is at least the straight line distance qqq ppp .

1.2.5 Find the arc-length of the curve

(t) = (3t2, t 3t3)

starting at t = 0.


1.2.6 Find, for 0 x , the arc-length of the segment of the curve

(t) = (2 cos t cos 2t, 2 sin t sin 2t)

corresponding to 0 t x.1.2.7 Calculate the arc-length along the cycloid in Exercise 1.1.7 corresponding to one

complete revolution of the circle.

1.2.8 Calculate the length of the part of the curve

(t) = (sinh t t, 3 cosh t)

cut off by the x-axis.

1.2.9 Show that a curve such that... = 0 everywhere is contained in a plane.

1.3.1 Which of the following curves are regular ?(i) (t) = (cos2 t, sin2 t) for t R.(ii) the same curve as in (i), but with 0 < t < /2.(iii) (t) = (t, cosh t) for t R.Find unit-speed reparametrizations of the regular curve(s).

1.3.2 The cissoid of Diocles (see below) is the curve whose equation in terms of polarcoordinates (r, ) is

r = sin tan , /2 < < /2.

Write down a parametrization of the cissoid using as a parameter and showthat

(t) =


t31 t2

), 1 < t < 1,

is a reparametrization of it.


1.3.3 The simplest type of singular point of a curve is an ordinary cusp: a point pppof , corresponding to a parameter value t0, say, is an ordinary cusp if (t0) = 0and the vectors (t0) and

... (t0) are linearly independent (in particular, these

vectors must both be non-zero). Show that:(i) the curve (t) = (tm, tn), where m and n are positive integers, has an ordinarycusp at the origin if and only if (m,n) = (2, 3) or (3, 2);(ii) the cissoid in Exercise 1.3.2 has an ordinary cusp at the origin;(iii) if has an ordinary cusp at a point ppp, so does any reparametrization of .

1.3.4 Show that:(i) if is a reparametrization of a curve , then is a reparametrization of ;(ii) if is a reparametrization of , and is a reparametrization of , then isa reparametrization of .

1.3.5 Repeat Exercise 1.3.1 for the following curves:(i) (t) = (t2, t3), t R.(ii) (t) = ((1 + cos t) cos t, (1 + cos t) sin t), < t < .

1.3.6 Show that the curve

(t) =



1 + t2

), t > 0,

is regular and that it is a reparametrization of the curve

(t) =

(2 cos t

1 + sin t, 1 + sin t


2< t 0 for all t, and s = s if d/dt < 0 for all t.

1.3.9 Suppose that all the tangent lines of a regular plane curve pass through somefixed point. Prove that the curve is part of a straight line. Prove the same resultif all the normal lines are parallel.

1.4.1 Show that the Cayley sextic

(t) = (cos3 t cos 3t, cos3 t sin 3t), t R,


is a closed curve which has exactly one self-intersection. What is its period? (Thename of this curve derives from the fact that its Cartesian equation involves apolynomial of degree six.)

1.4.2 Give an example to show that a reparametrization of a closed curve need not beclosed.

1.4.3 Show that if a curve is T1-periodic and T2-periodic, it is (k1T1 +k2T2)-periodicfor any integers k1, k2.

1.4.4 Let : R Rn be a curve and suppose that T0 is the smallest positive numbersuch that is T0-periodic. Prove that is T -periodic if and only if T = kT0 forsome integer k.

1.4.5 Suppose that a non-constant function : R R is T -periodic for some T 6= 0.This exercise shows that there is a smallest positive T0 such that is T0-periodic.The proof uses a little real analysis. Suppose for a contradiction that there is nosuch T0.(i) Show that there is a sequence T1, T2, T3, . . . such that T1 > T2 > T3 > > 0and that is Tr-periodic for all r 1.(ii) Show that the sequence {Tr} in (i) can be chosen so that Tr 0 as r .(iii) Show that the existence of a sequence {Tr} as in (i) such that Tr 0 asr implies that is constant.

1.4.6 Let : R Rn be a non-constant curve that is T -periodic for some T > 0.Show that is closed.

1.4.7 Show that the following curve is not closed and that it has exactly one self-intersection:

(t) =

(t2 3t2 + 1

,t(t2 3)t2 + 1


1.4.8 Show that the curve

(t) = ((2 + cos t) cost, (2 + cos t) sint, sin t),

where is a constant, is closed if and only if is a rational number. Show that,if = m/n where m and n are integers with no common factor, the period of is 2n.

1.5.1 Show that the curve C with Cartesian equation

y2 = x(1 x2)

is not connected. For what range of values of t is

(t) = (t,t t3)


a parametrization of C? What is the image of this parametrization?1.5.2 State an analogue of Theorem 1.5.1 for level curves in R3 given by f(x, y, z) =

g(x, y, z) = 0.

1.5.3 State and prove an analogue of Theorem 1.5.2 for curves in R3 (or even Rn).(This is easy.)

1.5.4 Show that the conchoid(x 1)2(x2 + y2) = x2